Solve Conjugate Complex Equations: Rules & Examples

[matsorz]

I have a couple of problems with the conjugate. I have two equations to solve, (conjugate)z=2/z and (conjugate)z=-2/z
How do I solve these= Are there some rules when you use the conjugate?

 

[tiny-tim]

hi matsorz!

$z + \bar{z} = 2Re(z)$

$z - \bar{z} = 2Im(z)$

$z\bar{z} = |z|^2$ ​

 

[matsorz]

Ok, so I put |z^2|=2 in a) and |z^2|=-2 in b? Do I put in x and y, or do i just solve for z straight away?

 

[HallsofIvy]

One problem you have is that the second equation is impossible.
$$\overline{z}= -\frac{2}{z}$$
is the same as $z\overline{z}= -2$ but $z\overline{z}= |z|^2$ must be a positive real number.

 

[stallionx]

I have a couple of problems with the conjugate. I have two equations to solve, (conjugate)z=2/z and (conjugate)z=-2/z
How do I solve these= Are there some rules when you use the conjugate?

(a-bi)*(a+bi)=2
a**2 + b**2 = 2

center ( 0,0 ) radius =sqrt2

(a-bi)*(a+bi)=-2

center (a,b) radius = i sqrt2

take abs of radii both

one is sqrt2
other abs ((sqrt 2) *i)=abs(i) * abs(sqrt2)

abs(i)=1

so they turn out to be equal.

 

[tiny-tim]

Ok, so I put |z^2|=2 in a) and |z^2|=-2 in b? Do I put in x and y, or do i just solve for z straight away?

straight away …

(it's always quickest to avoid coordinates as far as you can)

|z| = √2 (the first case), so the general solution for z is … ?